Real Analysis Exchange

Jensen’s Inequality for Conditional Expectations in Banach Spaces

August M. Zapała

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In this note we present a simple proof of the inequality $\Phi \left( E^{\mathcal{A}}\xi \right) \leq E^{\mathcal{A}}\Phi (\xi )$ a.s. for separable random elements $\xi \mathcal{I}n L_{1}(\Omega ,\mathcal{F},P;X)$ in a Banach space $X,$ where $E^{\mathcal{A}}\left(\cdot\right) $ denotes conditional expectation with respect to the $\sigma $-field $\mathcal{A} \subset \mathcal{F}$, and $\Phi :X\rightarrow \mathbb{R}$ is a convex functional satisfying certain additional assumptions which are less restrictive than known till now. Some consequences of the above result are also discussed; e.g., it is shown that if $\xi $ is a Gaussian random element in $X$, then there exists a constant $0<c< \infty $ such that for each $\sigma $-field $\mathcal{A}_{0}\subset \mathcal{F}$ the family $\left\{ \exp \{c\left\| E^{\mathcal{A}}\xi \right\| ^{2}\}\mathcal{A}_{0}\subseteq \mathcal{A} \subseteq \mathcal{F}\right\} $ is uniformly integrable.

Article information

Real Anal. Exchange, Volume 26, Number 2 (2000), 541-552.

First available in Project Euclid: 27 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D07: Inequalities involving other types of functions 60E15: Inequalities; stochastic orderings 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12] 60B11: Probability theory on linear topological spaces [See also 28C20]

conditional expectation Jensen's inequality convex function Gaussian random element


Zapała, August M. Jensen’s Inequality for Conditional Expectations in Banach Spaces. Real Anal. Exchange 26 (2000), no. 2, 541--552.

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